Problem 1
We need to ensure that the radicands (stuff under the square root) are never negative. So set each expression greater than or equal to 0. Then solve for x
For the first radicand, we get:
7-x >= 0
7-x+x >= 0+x
7 >= x
x <= 7
And the same story or the other radicand
x+2 >= 0
x+2-2 >= 0-2
x >= -2
So together x >= -2 and x <= 7
As one inequality, we'd write it as -2 <= x <= 7
However, if x = -2, then it leads g(x) to be zero. We cannot divide by zero. So we must exclude x = -2 from the domain.
The proper domain is -2 < x <= 7
So the answer is choice D
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Problem 2
You are correct. The domain is the set of real numbers with x = 0 kicked out of the domain. Basically, x can be any number but 0.